To solve for $P(X = 3)$, we need to determine the probability that the maximum number of consecutive heads in five tosses of a fair coin is exactly 3.

Step 1: Total Possible Outcomes

The total number of outcomes when tossing a fair coin 5 times is: $2^5 = 32$ Each sequence consists of heads (H) and tails (T).

Step 2: Count of Sequences where Maximum Consecutive Heads is 3

To satisfy $X = 3$, we need sequences where:

• There is at least one sequence of exactly 3 consecutive heads.
• No sequence of heads contains more than 3 consecutive heads.

Cases to Consider

We will count the valid sequences that have exactly 3 consecutive heads.

1. Three heads at the start: The sequence is $HHH$ followed by two additional flips. To ensure the maximum is exactly 3, at least one of the remaining flips must be tails.

   • Possible sequences: $HHH TT, HHH TH, HHH HT$
   • Count: 3

2. Three heads in the middle: The sequence is $THHH$ followed by one additional flip or one flip before $HHH$.

   • Possible sequences: $THHH T, THHH H$ or $T HHH T, H HHH T$ with exactly 3 heads.
   • Count: 4 (2 from each side)

3. Three heads at the end: The sequence starts with two flips followed by $HHH$. To ensure the maximum is exactly 3, at least one of the first two flips must be tails.

   • Possible sequences: $TT HHH, TH HHH, HT HHH$
   • Count: 3

Thus, the total number of valid sequences where the maximum number of consecutive heads is exactly 3 is: $3 + 4 + 3 = 10$

Step 3: Probability

Now, the probability is the ratio of favorable outcomes (10) to the total number of outcomes (32): $P(X = 3) = \frac{10}{32} = \frac{5}{16}$

Thus, the probability $P(X = 3)$ is $\frac{5}{16}$.

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Would you like further details or clarification on any step?

Here are 5 related questions for further exploration:

1. What is $P(X = 4)$?
2. How does the probability change if the number of coin tosses increases to 6?
3. How can you generalize this method to find $P(X = k)$ for any number of tosses?
4. What is the expected value of $X$ for 5 coin tosses?
5. What is the probability that there are no consecutive heads in 5 tosses?

Tip: In problems involving consecutive events (like heads in coin tosses), using pattern recognition and breaking the problem into smaller cases simplifies the solution.